相关论文: Prime twists of elliptic curves

Given an elliptic curve $E$ over $\mathbb{Q}$, a celebrated conjecture of Goldfeld asserts that a positive proportion of its quadratic twists should have analytic rank 0 (resp. 1). We show this conjecture holds whenever $E$ has a rational…

We establish a congruence formula between $p$-adic logarithms of Heegner points for two elliptic curves with the same mod $p$ Galois representation. As a first application, we use the congruence formula when $p=2$ to explicitly construct…

A weaker form of a 1979 conjecture of Goldfeld states that for every elliptic curve $E/\mathbb{Q}$, a positive proportion of its quadratic twists $E^{(d)}$ have rank 1. Using tools from Galois cohomology, we give criteria on E and d which…

For positive rank $r$ elliptic curves $E(\mathbb{Q})$, we employ ideal class pairings $$ E(\mathbb{Q})\times E_{-D}(\mathbb{Q}) \rightarrow \mathrm{CL}(-D), $$ for quadratic twists $E_{-D}(\mathbb{Q})$ with a suitable ``small $y$-height''…

Let $K$ be a number field and $E/K$ be an elliptic curve with no $2$-torsion points. In the present article we give lower and upper bounds for the $2$-Selmer rank of $E$ in terms of the $2$-torsion of a narrow class group of a certain cubic…

Given an elliptic curve E over the rational with no rational 2-torsion points, we prove the existence of a quadratic twist of E for which the 2-Selmer rank is less than or equal to 1. By the author's earlier result, we establish a lower…

Let $E$ be an elliptic curve over $\Bbb{Q}$ with the given Weierstrass equation $ y^2=x^3+ax+b$. If $D$ is a squarefree integer, then let $E^{(D)}$ denote the $D$-quadratic twist of $E$ that is given by $E^{(D)}: y^2=x^3+aD^2x+bD^3$. Let…

Extending the idea of \cite{dab2} and using the 2-descent method, we provide three general families of elliptic curves over Q such that a positive proportion of prime-twists of such elliptic curves have rank zero simultaneously.

Watkins' conjecture asserts that the rank of an elliptic curve is upper bounded by the $2$-adic valuation of its modular degree. We show that this conjecture is satisfied when $E$ is any quadratic twist of an elliptic curve with rational…

We study the $2$-Selmer ranks of elliptic curves. We prove that for an arbitrary elliptic curve $E$ over an arbitrary number field $K$, if the set $A_E$ of 2-Selmer ranks of quadratic twists of $E$ contains an integer $c$, it contains all…

Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's…

For every n there exists an elliptic curve E over the rational numbers and an n-dimensional subspace V of non-zero rationals modulo squares such that for all v in V, the quadratic twist of E by v has positive rank.

We observe that there are elliptic curves over number fields all of whose quadratic twists must have positive rank, assuming the Birch-Swinnerton-Dyer conjecture. We give a classification of such curves in terms of their local behaviour,…

This paper investigates which integers can appear as 2-Selmer ranks within the quadratic twist family of an elliptic curve E defined over a number field K with E(K)[2] = Z/2Z. We show that if E does not have a cyclic 4-isogeny defined over…

Given an elliptic curve $E/\mathbb{Q}$, it is a conjecture of Goldfeld that asymptotically half of its quadratic twists will have rank zero and half will have rank one. Nevertheless, higher rank twists do occur: subject to the parity…

Let $K=\Bbb Q(\sqrt{-q})$, where $q$ is a prime congruent to $3$ modulo $4$. Let $A=A(q)$ denote the Gross curve. Let $E=A^{(-\beta)}$ denote its quadratic twist, with $\beta=\sqrt{-q}$. The curve $E$ is defined over the Hilbert class field…

We investigate variations of Selmer ranks under quadratic twists satisfying the Heegner hypothesis. In particular, starting with an elliptic curve $E/\mathbb{Q}$ with partial $2$-torsion and a common relaxed Selmer group, we derive explicit…

Let $E$ be an elliptic curve over $\Q$ without complex multiplication, and which is not isogenous to a curve with non-trivial rational torsion. For each prime $p$ of good reduction, let $|E(\F_p)|$ be the order of the group of points of the…

We prove that, when all elliptic curves over $\mathbb{Q}$ are ordered by naive height, a positive proportion have both algebraic and analytic rank one. It follows that the average rank and the average analytic rank of elliptic curves are…

For an elliptic curve $E/\Q$, we determine the maximum number of twists $E^d/\Q$ it can have such that $E^d(\Q)_{tors}\supsetneq E(\Q)[2]$. We use these results to determine the number of distinct quadratic fields $K$ such that…